The distance matrix of a graph \(G\)  of order \(n\)  is the \(n \times n\)  matrix \(\mathcal {D}(G) = \left(d_{i,j}\right)\) , indexed by the vertices of \(G\) ,
where \(d_{i,j}=d(v_{i},v_{j})\) . Two of the oldest works on the distance matrix are [1]} (1935) and [2]} (1938). In [3]} (1964), the authors ask for the conditions under which a given a square symmetric matrix
with real and nonnegative entries is realized as the distance matrix of a graph. Different realizability problems of distance matrices are investigated in [4]}, [5]}, [6]}, [7]}, [8]}, [9]} and [10]}, among many others papers. A remarkable result on the distance matrix, due to Graham and Pollak [11]} (1971), is : If \(T\)  is a tree of order \(n \ge 2\)  with distance matrix \(\mathcal {D}(T)\) , then
\(\text{det} \mathcal {D}(T) = (-1)^{n-1}\left(n-1\right)2^{n-2}. \) 
